42 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA It should be carefully noted that, if we suppose only that u is injective (resp. surjective), then u is not necessarily injective (resp. surjective) (Prob- lem 9). (12.9.5) Let Gbea metrizabk group in which there exists a neighborhood V of e which is complete (with respect to some left- or right-invariant distance). Then G is complete. In particular, a locally compact metrizable group is complete. Let d be a left-invariant distance on G and let (xn) be a left Cauchy sequence in G. Let £ > 0 be such that the closed ball B'(e; s) is contained in V. By hypothesis there exists an integer n0 such that d(xn, xm) ^ e for all m^nQ and n^.n09 hence the sequence (xn)n^no is contained in the closed ball B'(*no; e). But this closed ball is a complete subspace, because it is obtained by left translation from B'(#; &) which is closed in Y (3.14.5). Hence the se- quence (xn)n>1 converges in G. The last assertion follows from (3.16.1). (12.9.6) In a Hausdorff topologicalgroup G, every locally compact metrizable subgroup H is closed. Let x e H, let V be a neighborhood of e in G such that V n H is compact, and let W be a symmetric neighborhood of e in G such that W2 c V. Then xW n H is nonempty and relatively compact in H, because if y0 e xW n H then for each y e xW n H we have yQ 1y e W2 n H cz V n H and therefore y e 7o(V n H), which is a compact set. It follows (12.3.6) that the closure of xW n H in G is contained in H, and therefore x e H. (12.9.7) Let G be a Hausdorif commutative topological group, written additively. All the material on series in Section 5.2 which involves only the topology of G remains valid without any change. The same is true of Cauchy's criterion (5.2.1) if G is metrizable, by replacing the norm ||x|| by rf(0, x), where d is an invariant distance on G. PROBLEMS 1. Let G = GL(2, R) be the multiplicative group of all real 2x2 nonsingular square matrices. For each integer n > 0, let Vn be the set of matrices X—1X y\ e G such that x- 1| <;i//i, \y\ <S l/it,>| g 1/n, and |f- 1| 5 l/n. Show that the family of sets Vn is a fundamental system of neighborhoods of the neutral element / of G for a topology f compatible with the group structure of G (cf. Section 12.8, Problem 1). The group G is locally compact in the topology ^", but &~ cannot be defined by antity mappings on G! and